Method and apparatus for determining the angular momentum vector of a satellite

Abstract

The invention relates to a method for determining the angular momentum vector of a satellite, located in an external magnetic field, relative to a satellite-integral orthogonal system of coordinates. The satellite has a torque generating system to generate pulse-shaped torques around all three axes. A magnetometer measuring on three axes is used, whose measured values are fed to a computer. In the latter, at successive points in time and using the current measured values, an observer equation is integrated, with new estimated values being determined for the angular momentum vector. Using these as well as the current measured values, an inequality is continuously checked; when it is not fulfilled, the torque generating system operates briefly, so that finally the fulfillment of the inequality is forced and identity of the last estimated value that has been determined with the rotary pulse vector to be determined is achieved.

What is claimed is: 
     
       1. Method for determining an angular momentum vector H of an object located in an externally generated magnetic field B, relative to an orthogonal system of coordinates whose axes x,y,z coincide with main inertia axes of the object, said object having a magnetometer for measuring magnetic field strength in the direction of said x, y and z axes, and a torque generating system for producing torque pulses about all three axes, said method comprising the steps of: measuring magnetic field strength B of said externally generated magnetic field along said x, y and z axes by means of said magnetometer, wherein B=(B i ) T  (i=x 1  y,z);   entering measured values of said magnetic field strength B into a computer;   said computer calculating estimated values H of angular momentum of said object at successive points in time, for each calculation using most recently measured values of said magnetic field strength and integrating the observer equation   H=H×I.sup.-1 H+I.sup.-1 A.sup.T K(Y-AI.sup.-1 H)+M     with ##EQU20## wherein M is an estimated value for an angular torque vector; I is an inertia matrix of the satellite, and I -1  is its inverse matrix; A is a quadratic matrix of rank 3 with three mutually orthogonal line vectors, one of which coincides with a normalized magnetic field vector b≡B/|B|, and A T  is its transposed matrix; K is a quadratic starting matrix K=(k lm ) with rank 3 (l,m=1,2,3), with coefficients k lm  equal to zero in a column whose number coincides with a number of the line in matrix A in which the normalized magnetic field vector b is located, and whose other coefficients must be determined in such fashion that the matrix ##EQU21## is positively definite or semi-definite; with ##EQU22## wherein I x ,y,z represent diagonal components of inertia matrix I; continuously checking calculated values H and current measured values of said normalized magnetic field vector b to determine whether an inequality     (Ib×b).sup.T H≧0     is satisfied;     if said inequality is not satisfied, said torque generating system applying a torque M=(M x ,M y ,M z ) T  to said object for time intervals Δt x , Δt y , Δt z , to cause a change in angular momentum ΔH=(M i  Δt i ) T  whereby said inequality is satisfied; and   performing successive integrations until such integrations no longer result in a change in |H|, whereby the calculated value of H is equal to an actual angular momentum vector H of said object.   
     
     
       2. Method according to claim 1 wherein the following relationship is created between the elements of matrix K≡(k lm ) and those of matrix N≡(n lm ): ##EQU23## where c 1 ,2 >0, so that: ##EQU24## 
     
     
       3. Method according to claim 1 wherein time intervals Δt i  according to the requirement ##EQU25## are determined, with the following still being valid:   sign M.sub.i =sign (Ib)×b!.sub.i =-signH.sub.i.     
     
     
       4. Method according to claim 1 wherein two line vectors of matrix A that do not coincide with the normalized magnetic field vector b in the case |b z  |=1 become   a.sub.1 =(1,0,0).sup.T, a.sub.2 =(0,1,0).sup.T sign b.sub.z     and in the case |b z  |<1 become ##EQU26## with the following for A: ##EQU27##   
     
     
       5. Method for determining an angular momentum vector H of an object located in an externally generated magnetic field B, relative to an orthogonal system of coordinates whose axes x,y,z coincide with main inertia axes of the object, said object having a magnetometer for measuring magnetic field strength on said x, y and z axes, and a torque generating system for producing torque pulses about all three axes, said method comprising the steps of: measuring magnetic field strength B of said externally generated magnetic field along said x, y and z axes by means of said magnetometers, wherein B=(B i ) T  (i=x 1  y,z);   calculating estimated values H of angular momentum of said object at successive points in time using measured values of magnetic field strength according to a predetermined formula using said measured values of magnetic field strength;   after each calculation of H, using a most recently calculated value of H, current measured values of said magnetic field strength and inertia parameters of said object to determine whether a predetermined convergence criterion is satisfied;   if said predetermined convergence criterion is not satisfied, said torque generating system applying incremental torque impulses to said object on said x, y and z axes, said impulses having a magnitude and duration such that said convergence criterion is satisfied; and   repeating said process until successive calculations result in no change in said estimated value of angular momentum, whereby said estimated value of angular momentum equals actual angular momentum of said object.   
     
     
       6. Apparatus for determining an angular momentum vector H of a satellite located in an externally generated magnetic field, relative to an orthogonal system of coordinates whose axes x, y and z coincide with main inertia axes of the object, said apparatus comprising: a magnetometer for measuring magnetic field strength of said externally generated magnetic field along said x, y and z axes;   a computer coupled to receive measured values of magnetic field strength from said magnetometer, for calculating estimated values H of angular momentum of said satellite at successive points in time according to a predetermined formula using said measured values of magnetic field strength;   means responsive to each calculation of an estimated value of angular momentum, for determining whether a predetermined convergence criterion is satisfied, based on a most recently calculated value of angular momentum, current measured values of said magnetic field strength and inertia parameters of said satellite;   means operative when said convergence criterion is not satisfied for applying incremental torque impulses to said satellite on said x, y and z axes, said impulses having a magnitude and duration such that said convergence criterion is satisfied; and   means for reading said estimated value of angular momentum as an actual value of angular momentum of said satellite when successive calculations of said estimated angular momentum H produce no change therein.